Basis

A set of vectors in a vector space is called a basis for if:

spans .
is linearly independent.

Example

Warning

What the fuck is this?

Warning

What the fuck is this pt.2

A few theorems

Theorem 1

If is a basis for a vector space , then every set containing more than vectors is linearly dependent.

has two basis vectors, while has 3.

Theorem 2

If a vector space has one basis with vectors, then every other basis for should have vectors.

Dimension of Vector Space

If a vector space has a basis consisting of vectors, then the number is called the dimension of and denoted by

COMPLETE PAGE 4-6 in Basis paper

How to find a basis for a matrix

Row space

If we wanted to find the Row Space of a given matrix , we would need to use elementary row operations to convert into row echelon form.

The non-zero row vectors in R.E.F. form a basis for the row space of .

Example

Column space

We either do the same thing with , or:

We take the R.E.F. again, then the columns of matrix B, which are Linearly independent because they have leading 1's, are also Linearly independent in A, and these columns in A form the basis of the column space of A.

Example

Note

This method doesn't necessarily give the same basis that results from the Transpose method.

How to find basis for a subspace

The technique above can be also used for this.
We have the set that spans a subspace in .
The idea is to use the vectors in as the rows of a matrix , then rewrite in R.E.F.

Example